Identify the mathematical concept chosen, and briefly describe its real-life application.

Each of these topics has a wide variety of real-life applications. For your culminating activity, your task is to choose a topic from this course, and examine one of its applications in the real world. The topic you choose is up to you, and the application you examine is also up to you. Some possible examples are
listed at the bottom of this page. Your completed project should include the following elements:
1) Title Page

2) Clear statement of topic that is being examined
• Identify the mathematical concept chosen, and briefly describe its real-life application.

3) Background Research/Prior Knowledge
• Research your topic and include as much information as you reasonably can about your topic. Begin by summarizing what we have learned in this course about the topic, and then explain how it applies to the area that you have chosen to examine. You will need to cite your sources accurately here (i.e., page #s of textbook), and be sure that you choose reliable sources of information.

4) Sample Problem and Solution
• Create a sample word problem (including an equation) from the area that you have chosen to examine, and include a full, detailed solution of the problem. Your problem and solution should be of a complexity level appropriate to the level of this course and be solved using a method taught in class. Refer to the appropriate Chapter in your textbook or class notes for ideas on application type questions.

5) Conclusion
• Summarize what you have learned about the application you have chosen. Discuss any things which you are still not sure about, or what you would like to learn next about this topic.

6) References
• A complete list of all sources that you have consulted in completing this project. You may use textbooks, websites, and other reliable sources of information. Your list of references should be in APA format.

Examples of Topics:
• Polynomial Functions and Architecture
• Rates of Change and Kinematics
• Trigonometric Functions and Electromagnetic Waves
• Logarithmic Functions and Seismic Activity (i.e., the Richter Scale)
• Exponential Functions and Population Growth
• Any number of others…choose something that is of interest to you!

Determine the set of solutions of the new system depending on the parameter k.

5. (7 points) Consider the following linear system S 2w + x + y +z = a w+y+z= b W+y= 2 (a) (3 points) Determine the dimension of the space of the solutions and the set of solutions of system S employing Gaussian elimination. (b) (4 points) Assume that a fourth equation w + ky 1, where k is a real parameter, is added to the system. Determine the set of solutions of the new system depending on the parameter k.

Compute two distinct, positive integer solutions to x2 − 17y2 = 1.

MA2011/21
Section A
1. (a) Use Fermat’s Method to factorize 1147. [5]
(b) Euclid’s algorithm applied to two numbers a and b computed the quotients q1 = 3,
q2 = 3, q3 = 3, q4 = 3, q5 = 3 (in this order) and their greatest common divisor
gcd(a, b) = 3. Compute a and b. [5]
2. Using the extended version of Euclid’s algorithm, find a solution to the following Dio-
phantine linear equation.
77x + 91y + 143z = 2 [10]
3. Find all solutions for the following pair of simultaneous congruences.
262x ≡ 3 mod 807
3x ≡ 2 mod 5 [10]
4. Show that the equation
2×3 + 7y3 = 4
has no solution in integers. [10]
5. (a) Derive the continued fraction of √7. [5]
(b) Find the value of β, given its continued fraction expression β = [1, ̄7], i.e., a0 = 1
and ai = 7 for all i ∈ {1, 2, . . .}. [5]
– 2 –
MA2011/21
Section B
6. The Euler’s function φ(m) counts the number of integers a with 0 ≤ a < m and gcd(a, m) = 1. (a) Let m = pa where p is prime. Show that φ(m) = m(1 − 1/p). [5] (b) Let m = pa and n = qb where p and q are distinct primes. Show that φ(mn) = φ(m)φ(n). [5] (c) Compute φ(17) and φ(77). [4] (d) Show that if gcd(a, m) = 1, then aφ(m) ≡ 1 mod m. [11] 7. (a) Show that there exists a constant c > 0 such that
|b√7 − a| ≥ 1
cb ,
for all natural numbers a, b (b 6 = 0). [7]
(b) Compute two distinct, positive integer solutions to x2 − 17y2 = 1. [7]
(c) Let α > 0 be a real number and let pn/qn denote the corresponding n-th conver-
gent.
Show that ∣
∣
∣
∣
pn
qn
− α
∣
∣
∣
∣ ≥ 1
2q2
n
, and
∣
∣
∣
∣
pn+1
qn+1
− α
∣
∣
∣
∣ ≥ 1
2q2
n+1
,
implies (qn+1 − qn)2 ≤ 0.
Conclude that either
∣
∣
∣ pn
qn − α
∣
∣
∣ < 1
2q2
n
or
∣
∣
∣ pn+1
qn+1 − α
∣
∣
∣ < 1
2q2
n+1
. [11]
– 3X

iscuss an appropriate due date with your teacher and omplete your project in the agreed timeline.

Over the last three units, you have learned about numbers of outcomes and how to calculate permutations and combinations. Read the example of a children’s story below. Then use the example to create your own story that centres around permutations and combinations. Present your story and the answer to the numbered questions below in a Google Slide.
Goldilocks Example
Goldilocks Example
Goldilocks was walking in the woods and stumbled upon a cabin. Upon entering the cabin she smelled something cooking on the stove. When she approached the stove, she sees porridge cooking in three different pots (red, green, and blue). On the counter, she sees three different sized bowls (small, medium, and large). She realizes there are so many options for her to have some porridge that she can’t think straight and creates the diagram on the left to count them. NINE! That’s a lot of choices! She realizes she could have multiplied the 3 choices by the 3 other choices and had the right answer!

⦁ Create your own children’s story (or use one that already exists as a starting point) that involves permutations, combinations, or Pascal’s Triangle and the character is facing a decision that involves large numbers. Carry the problem through the story and use a diagram to model the situation so that the character can help figure out the answer. Do not use combinatorial notation in the problem (DON’T USE THIS – 8 C 6 or 5 P 3 ).
⦁ Write a summary of the math involved in your story using combinatorial notation (YOU SHOULD USE THIS – 8 C 6 or 5 P 3 ). You can incude this math in your slideshow as pictures of your handwritten math (it does not need to be typed).
⦁ Put a copy of your completed project in your course folder and notify your teacher.

Discuss an appropriate due date with your teacher and omplete your project in the agreed timeline. Read the rubric on the next page to see how you will be graded.

Combinations Unit Rubric
Student:
Level 1 = No Level 2 = Somewhat Level 3 = Good Level 4 = Excellent
Categories Questions Grade
K/U Did you demonstrate knowledge of combinations and permutations?

Did you demonstrate knowledge of combinatorial notation?
T/I Did you write a creative story involving large numbers?

Did you draw a correct diagram to help out your main character?
C Did you communicate the story and problem well?

Did you write out the math showing steps and communicating your thinking?
A Did you apply your knowledge to complete a correct solution and apply it to a real-life (your children’s’ story) situation?

Did you demonstrate an ability to put all of this information into a professional-looking Google Slide?

Identify the potential hazards arising from the work activity under the current lighting conditions in the workplace.

K 4: Lighting Measurement Lab
(15 marks]

Light Measurement Laboratory, illuminance measurements were performed of a certain lab space. Write a aballaboratory report, covering the following topics: a. Laboratory Results b. Laboratory Analytical Calculations and Discussion Your analysis and discussion should also cover the following topics: Identify the potential hazards arising from the work activity under the current lighting conditions in the workplace; Discuss who may be harmed; Evaluate risks and decide whether improvement measures are needed to protect the users, including but not limited to the lighting provision. Think of the lamp fixtures and types, and provide your best recommendation options (at least two). Include the information related to the Assessment Record Form. You can use diagrams, graphs, calculations, tables, sketches, photographs, and external sources to support your answer.

Physics of Light & Lighting Design

[110 marks]

A reading lamp utilises a 70 W GLS light bulb. Assuming the lamp emits light equally: (a) Provide an estimate of the flux and intensity, in appropriate units. (b) What is the illuminance at a distance 1.5m from the lamp? Is it enough for comfortable reading?

Impact of a Jet Laboratory

[15 marks]

A circular jet hits a disc normally, at a flow rate Q= 5.10’3 m3/s. The plate used is hemispherical (i.e. 180′ deflection), the nozzle diameter is d = 60 mm, and the density of water is p = 1 kg/I. Determine the following parameters, making unit conversions where appropriate: i) Cross sectional area, A ii) Jet velocity, u iii) The jet force, F, on the plate iv) Now assume that you have a flat (0 degrees) and a 30 degree plate. What would the force be? Calculate the forces appropriately. Discuss the differences in forces related to the different shapes.

Health & Safety

[10 marks]

You are working as a construction manager for the design and construction of a building and you must complete a risk assessment form, identifying the possible hazards that you might expect in this project. You need to address all aspects of a building covered in this module, e.g. heat transfer and thermal control, physics of light, ventilation and humidity control, physics of sound and acoustic design, physics of water and hydraulic design. Furthermore, you need to investigate a minimum of two design management regulations and describe those.

 

Calculate the maximum shear force and state its location along the beam.

Science and materials

The word count is 1,500 words excluding formulas, calculations, tables, figures, captions, and citations/references.

SECTION 1: MATERIALS TASK I:

Structural Analysis

The simply supported steel beam in FIGURE 1 has a span of 4m and carries a total marks] 110 Uniformly Distributed Load (UDL) of q = 30 kN/m, including the self-weight.

For this beam:

Calculate the reactions at the supports, showing your calculations in full.

Draw the Shear Force and Bending Moment diagrams.

Calculate the maximum shear force and state its location along the beam.

Calculate the maximum bending moment and state its location along the beam.

How long will the activity take when implemented in the classroom?

Lesson plan and presentation at an Elementary math level about Stem Leaf Plots

Include the following:

Slide 1&2: Lesson Plan: Stem Leaf Plots

Slide 1:

Overview: Write an introduction to the class activity. Include the purpose of the activity and desired outcome.

Objectives: The objectives should be specific and measurable.

Time: How long will the activity take when implemented in the classroom?

Materials: Describe any materials that are needed to conduct the lesson.

Slide 2:

Activity: Provide a detailed description of the activity. Write all steps from the instruction of the assessment.

Slides 3-7:Presentation for lesson plan

1. Presentation: Complete a PowerPoint presentation that could be used in class to teach the lesson plan.

2. Notes Section: The PowerPoint must include presentation notes at the bottom of each slide.

Instructions and scoring guide in the attached file.

Determine the linear correlation and regression equation between two variables to make predictions for the dependent variable.

Assignment Content

Competencies

⦁ Describe the data using the measures of central tendency and measures of variability.
⦁ Apply the normal distribution, standard normal distribution, and central limit theorem.
⦁ Develop a confidence interval for a population parameter.
⦁ Evaluate hypothesis tests for population parameters from one population.
⦁ Evaluate hypothesis tests for population parameters from two populations.
⦁ Determine the linear correlation and regression equation between two variables to make predictions for the dependent variable.